reserve L for Lattice;
reserve X,Y,Z,V for Element of L;

theorem
  X = Y "\/" Z iff Y [= X & Z [= X & for V st Y [= V & Z [= V holds X [= V
proof
  thus X = Y "\/" Z implies Y [= X & Z [= X & for V st Y [= V & Z [= V holds X
  [= V by FILTER_0:6,LATTICES:5;
  assume that
A1: Y [= X & Z [= X and
A2: for V st Y [= V & Z [= V holds X [= V;
A3: Y "\/" Z [= X by A1,FILTER_0:6;
  Y [= Y "\/" Z & Z [= Y "\/" Z by LATTICES:5;
  then X [= Y "\/" Z by A2;
  hence thesis by A3;
end;
